On <span class="mathjax-formula">$k$</span>-very ampleness of tensor products of ample line bundles on abelian surfaces

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

Y ^OSHIAKI F ^UKUMA

On k-very ampleness of tensor products of ample line bundles on abelian surfaces

Rendiconti del Seminario Matematico della Università di Padova, tome 101 (1999), p. 209-227

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of Ample Line Bundles

on

Abelian Surfaces (*).

YOSHIAKI

FUKUMA (**)

ABSTRACT - In this _paper, we

study

abelian surfaces X and

ample

line bundles

L1,

^{... ,}

L,

^{such that L}

:= L1

^{+ ...}

+ Lt

^{is not}

k-very ample

^{for t =} k, ^{k + 1. And}

we also

study polarized

abelian surfaces (X, L) ^{such that} ( k - t ) L ^{is not k-}

very

ample

^{with t ~ 1 under some} condition. As corollaries of the above re-

sults, ^we get ^the classification of (X, L ) ^{such that} ( k - t ) ^{L is not}

k-very ample

fort=-1,0,1 ^{and 2.}

0. Introduction.

Let X be an abelian

variety

^{over the}

complex

^number field C and let L be an

ample

line bundle on X. Then it is well-known that 2L is

spanned

and 3L is _very

ample. (See [LB].)

^In

[BaSzl]

^and

[BaSz2],

^{Bauer and}

Szemberg

^{studied a} sufficient condition of

k-very ampleness

^of

L1

^{+ ... +}

+

Lk

+ 1 for

ample

line bundles

Ll ,

^{... ,}

Lk

+ 1. In

particular,

ⁱⁿ

[BaSz2],

^{as a}

corollary, they proved

^that

L1

^{+ ... +}

Lk

^{+ 2 is}

k-very ample

^for any

ample

line bundles

L1,

^{... ,}

Lk

^{+ 2 .}

Next the

following question

arises from the

Bauer-Szemberg

^re-

sult ;

(*) Research Fellow of the

Japan Society

^{for the} Promotion of Science.

(**) ^Indirizzo dell’A.: E-mail:

[email protected].

Current Address:

Department

^of Mathematics,

College

^of Education, ^Naruto

University

^{of Educa-} tion, Takashima, Naruto-cho, Naruto-shi 772-8502,

Japan;

^E-mail: fukuma@na-

ruto-u. ac. j p

QUESTION.

^{Let X be an abelian}

variety

^{X and let}

L1,

^{... ,}

L,

^be

ample

line bundles on X for s ~ k ⁺ 1. Then

classify (X, Li , ... , Ls )

^{such that}

L1

^{+ ... +}

L,

^{is not}

k-very ample.

Ohbuchi

([0])

studied

polarized

^{abelian varieties}

(X, L)

such that 2L is _{not very}

ample.

In sect.

2,

^we

study

^abelian surfaces X and

ample

^{line bundles}

L1, ... , Lt

^{such that}

L : = L1 + ... + Lt

^{is not}

k-very ample

^for

t = k,

k+1.

In sect.

3,

^we

study polarized

abelian surfaces

(X, L)

such that

(k -

-

t)

^{L is} not

k-very ample

with t &#x3E; 1 under some condition. In

particular,

we characterize

(X, L)

^{such that}

( k - t)L

^is not

k-very ample

^{with t =1}

or 2.

(See

Theorem 3.4 and Theorem

3.5.)

We work over the

complex

number field and we use the

customary

notation in

algebraic geometry.

Acknowledgment.

The author would like to express his

hearty

^thanks

to Professor Tomasz

Szemberg

^for

giving

^{him some useful comments}

and

suggestions.

The author also would like to thank the referee for

giv- ing

him useful comments.

1. Preliminaries.

DEFINITION 1.1.

(See [BeSo].) -

^Let

(X, L)

^{be a}

polarized

^surface.

Then L is called

k-very ample

^{if for} any 0-dimensional subscheme

(Z,

^with

length

c9z ^{= k +}

1,

^the map

is

surjective.

LEMMA 1.2. Let X be an abelian

surface.

Assume that X contains a

smooth

elliptic

^curve D. Then there exists an

elliptic fibration f:

^{X ~ C}

such that C is a smooth

elliptic

^{curve, D is a}

fiber of f,

^and any

fiber of f

is

isomorphic

^{to D.}

PROOF.

By

^a translation of

D,

^we may ^assume that D contains the

origin

of X and D is an abelian

subvariety

^{of X. Then} there exist the _quo- tient

XID

^{and the}

surjective homomorphism f:

^Then

XjD

^{is a}

smooth

elliptic

^{curve and} every fiber

of f is isomorphic

^{to the fiber over}

the

origin

^of

XID

which is D

by

construction.

LEMMA 1.3. Let X be an abelian

surface.

Assume that there exist a

smooth

elliptic

^{curve C and a}

suryective mor~phism f :

^{X -~ C with con-}

nected

fibers

^{such that}

f

^{has a section} S. Then X ⁼ C ^x F and

f

^{is identi-}

fied

^{with the}

first projection

^{via this}

isomorphism,

^{and S is a}

fiber of

the second

projection,

where F is ^a

fiber of f.

PROOF. We remark

that f is

^an

elliptic

fibration such that _any fiber of

f is

smooth since X is an abelian surface. Let S be a section

of f.

^Then

by

Lemma

1.2,

there exist a smooth

elliptic

^{curve C’ and an}

elliptic

^fibration

h : X ~ C’ ^‘ such that _any fiber of h is a smooth

elliptic

^{curve and S is a}

fiber of h. Moreover _any fiber of h

(resp. f)

^{is a} section of

f (resp. ^h).

^In

particular

C’ = F. Then there exists a

morphism

^such

that and where _P,

(resp. P2)

^{is the}

projection

C x C’ 2013&#x3E;C

(resp.

^C

x C" 2013&#x3E;C’).

^We remark that it is

bijective by

^con-

struction. Let

F f = f * ( x )

^and

Fh = h * ( y ),

^{where and}

Y E C ’.

^Then

and Then

Hence a is birational. Therefore

by

Zariski Main

Theorem,

^{we obtain}

that .7r is

an,isomorphism. 0

REMARK 1.3.1. Let

(X, L)

^{be a}

polarized

^abelian

^surface,

^{and let D}

be an effective divisor on X such that LD ⁼ 1. Then D is irreducible and reduced.

By Hodge

^index

Theorem,

^we

get

^that

^{D 2}

^{= 0.} Moreover D is a

smooth

elliptic

^{curve since X is an abelian surface and}

g(D)

^{= 1.}

By

^Lem-

ma

1.2,

there exists a fiber

space f :

X- C such that C is a smooth

elliptic

curve, D is a fiber of

f,

^and any fiber of

f

^is

isomorphic

^{to D. Since}

h ° (L)

&#x3E; 0 and LD =

1,

there exists a section

of f.

^So

by

^Lemma

1.3,

X = C ^x F for a fiber F

of f, and f is

identified with the first

projection

^via

this

isomorphism.

THEOREM 1.4

(Terakawa).

^Let

(X, L)

^{be a}

polarized

^{abelian sur-}

face.

^{Then L is}

ample if

^and

only if ^{L 2 ~}

^{4 k +} 6 and there exists

no

effective

^{divisor D}

satisfying

^the

inequalities

PROOF. See Theorem 3.15 in

[Te].

THEOREM 1.5. Let

(X, L)

^{be a}

polarized

^abelian

surface. If g(L) =

=

2,

^then

(X, L)

^{is one}

of

^the

following:

(I)

^{X =}

J(B),

^{and L} is the class

of

^a translation

of

the theta divi- sor, where B is a smooth

projective

^curve

of

genus

two, J(B)

^{is the}

jaco-

bian

variety of

^B.

(II)

^{X =}

El

^x

E2 ,

^{and L =}

P1*

6D, Q9

P2* ~2 ,

^where

E1 and E2

^are

smooth

elliptic

curves, pi:

E1

^x

E2 --~ Ei

is the i-th

projection,

^{and E}

E Pic

(Ei )

^with

deg

(J)1 =

deg

^=1.

PROOF. See

[OoU], [BeLP],

^or

[Fil. m

THEOREM 1.6. Let

(X, L)

^{be a}

polarized

^abelian

surface

^with

^{L 2 ;}

; 2 _a, where a E l~. Assume that there exists an irreducible reduced curve

D on X such that LD ⁼ 2 and

D 2 =

0. Then

(X, L) satisfies

^one

of

^the

following:

(1 )

^{X =}

E1

^x

E2

^{and L =}

pt

~31

(9 P2*

IB2 with

deg

$1 ~ ^{1 and}

deg B2

⁼

= 2,

^or

deg B1 = 2

^and

(2)

^{There exists a}

surjective morphism f :

X -~ C with connected

fibers

such that C is a smooth

elliptic

^{curve and} any

fiber of f

^{is a}

smooth

elliptic

^{curve, and}

^{(X, L)}

^{is one}

of

^the

following:

(2-1)

There exists an

ample spanned

^{vector bundle 8}

of

^{rank two}

on C such that 8 = 8’ Q9 J[1 ^with

deg

J[1 ~

~a/2 ~ ,

^{X is a double}

covering of IE~( ~ )

^whose branch locus is smooth and

linearly equivalent

^to

- 2 I~~~ ~ y f =

^{and L =}

C9(T) Of*

(2-2)

^{X =}

J(B)

^{and L =}

(9x(A) Of* ~1~

such that A is a transla- tion

of

the theta

divisor,

^{AF = 2}

for

^a

fiber

^F

of f,

^and

deg

~

~(a -1 )/2 ~ ,

(2-3)

^{X =}

E1

^X

E2

^{and L =}

P1*

6D, Q9

P2* 6D2 Q9f* ~

such that _any

fiber of

Pi is a section

of f for

ⁱ

= 1, 2, deg

^{(J)1 =}

deg (J)2

⁼

1,

^and

deg ME3 -&#x3E; ~(a - 1 )/2 ~ ,

where in

(1)

^and

(2-3) E1

^and

E2

^{are smooth}

elliptic

^curves, pi:

E1

^x

x

E2 - Ei

is the i-th

projection, Bi, Di

^{E Pic}

(Ei) for

^{i =}

1, 2,

^{E Pic}

( C),

in

(2-1)

^{8’ =}

oc for

21 ^{E Pic}

(C)

^with

oc

^and

2.el -= (9c, 3111

^E

E Pic ( C),

p:

I~( ~ )

-~ C is the natural

projection,

^:Jl:

^{X ~ P( 8)}

^{is the}

double

covering,

^{and T is a smooth}

elliptic

^{curve with TF = 2}

for

a

fiber

^F

of f,

^{and in}

^(2-2)

^{B is a smooth}

projective

^curve

of

genus

two, J(B)

^{is the}

jacobian variety,

^{and E Pic}

(C). (For

^{x E}

R, ~x]

denotes the smallest

integer

^{which is}

greater

^{than or}

equal

^to

x.)

REMARK 1.6.1. In the case

(2-2)

in Theorem

1.6,

^{B cannot be} gener- al.

Actually

^{if B is}

general,

then the Neron-Severi _group NS

(X)

^is gener- ated

by

the class of the theta divisor. In

particular,

^{X does not contain}

any

elliptic

^curve.

PROOF. This can be

proved by

^{the same}

argument

^{as in the}

proof

^of

Theorem 2.2 in

[Fk]. By assumption,

^{D is a smooth}

elliptic

^{curve. Hence}

by

^Lemma

1.2,

there exists a

surjective morphism f :

^{X- C} such that C is

a smooth

elliptic

^{curve, D is a fiber}

of f,

^and any fiber

of f is

smooth. Since LF ⁼ LD ⁼ 2 for _any fiber F

of f,

^we

get that f * o f * ^{(L )}

^{-~ L is}

surjective.

Let 8 :

= f * L.

^{Then 8 is a}

locally

free sheaf of rank two on

C,

^{and there}

exists a

morphism : X -~ lh( ~ )

^such

where : P(8)

^{- C is} the bundle _map.

By construction,

^{Jt is a double}

covering.

^{Since X is an}

abelian

surface,

the branch locus B is smooth and

linearly equivalent

^to

- 2

Kp(,).

Furthermore L = Jt *

(H(8))

^{and 8 is}

ample

^with a, where

H(8)

^{is the}

tautological

line bundle on

P(8). Since I - I

^has

a smooth

member, by

^{the same}

argument

^{as in the}

proof

^of

Proposition

2.3 in

[Fkl],

^{we can} prove that there exists a vector bundle 8’ on C and a

line bundle J1 on C such that 8 ⁼ 8’

(9 M,

^{and 8’ and 311}

satisfy

^{one of the}

following

^three

types;

(G~

there exists a nontrivial extension

and

where c4EPic(C)

^with

Let

c : IE~( ~ ) -~ Ih( ~’ )

^{be the}

isomorphism

such that Let JT

(1)

^{The case} in which 8’ is the

type (A)

^or

(B).

Let

Co

^E

[ H( 8’ ) [

^Then

^Co

^{is an} irreducible reduced curve

by Proposi-

tion 2.8 in

[Ha,

^Ch.

V].

^{Let B be} the branch locus of We remark that

-

2Kp(8’)

&#x3E; =

4 H( ~, ).

Since -2Kp(8’)CO=0,

^we

get Co c B

^or

Co n B = 0.

(1-1)

^{The case in which}

Co c B.

Then ⁼

2 B1.

^{Since LF = 2 for a fiber}

F of f

^and

B1

^{is not} contained in a fiber

of f, B1

^{is a section}

of f.

^Hence

by

^Lemma

^1.3,

^{X =}

= C ^x F and

f

^is identified with the first

projection,

^and

B1

^{is a fiber of}

the second

projection

h : C x F ~ F. So we

get

where 1P e Pic

(F)

^with

deg 1P

⁼ 2. Therefore we

get

^the

type (1)

^{in Theo-}

rem 1.6.

(1-2)

^{The case in which}

Co

^{n B = 0. Then we obtain one of the}

following:

where Bi

^{is an} irreducible reduced ^curve for

i = 2, 3,

^{4 with} First we consider the case

(1-2-1).

PROOF. We remark that

B2

^{is a smooth}

elliptic

^{curve. Hence}

by

^Lem-

ma

1.2,

there exists a

surjective morphism fl :

^{X -}

C1

^{such that}

C1

^{is a}

smooth

elliptic

^{curve and}

B2

^{is a fiber}

of fl .

^Hence

=

h ° (B2 )

^{= 1.} On the other

hand,

since 7r’ is a double

covering,

^{we have}

h ° ((,~’ ) * ( C° ) ) = h ° (H( ~’ ) ).

^Hence

~(~))=1.

Therefore ~’ is the

type ^(B).

^This

completes

^the

proof

of Claim 1.7.

Since L ⁼

(jr’)*(Co)0/*~

^we obtain the

type (2-1)

ⁱⁿ Theorem 1.6.

We remark that

B2 F

^{= 2 for a fiber}

F of f.

Next we consider the case

(1-2-2).

If ⁼

B3

⁺

B4 ,

^{then we}

get

^{= 0 and = 0. Since}

B3

^and

B4

^{are not} contained in a fiber

of f,

^we

get

^that

B3 F

⁼

B4 F

^{= 1 for}

a fiber F

of f because

^{= 2. Hence}

B3

^and

B4

^{are sections}

of f.

Hence

by

^Lemma

1.3,

^we

get

^{that X = C x}

F, f

^is identified with the first

projection,

^and

Bi

^{is a} fiber of the second

projection

h : X ~ F for

i = 3, 4.

Hence

by

^{the same}

argument

^{as the case}

(1-1),

^{we obtain L}

f* X 0

® h * ~’,

^{where 0’ E}

Pic (F)

^with

deg

^{1P’ = 2.}

Therefore we

get

^the

type ⁽¹⁾

ⁱⁿ Theorem 1.6.

(2)

^{The case in} which 8’ is the

type ^(C).

Then

H(8’)

^is

ample

^with

H(8’ )2

^{= 1. We}

put ox(A) = (Jt ’ )* (H(8’)).

Then AF ⁼ 2 for a fiber

F of f.

^Since

^{A 2 =}

^{2 and X is an abelian}

surface,

we obtain

g(A)

^{= 2 and we}

get

^that

(X, A)

^{is one of the}

type (I)

^or

(II)

ⁱⁿ Theorem 1.5. We remark that L ⁼

(jr’ )* (H(~’ ) ) ® f *

^{J1 =}

Of*

If

(X, A)

^{is the}

type (II)

^{in Theorem}

1.5,

^then any fiber

of f is

^{not con-}

tained in a fiber of _pi and _any fiber of _pi is a section

of f for

ⁱ

= 1,

^{2 since} AF ⁼ 2. Therefore if

(X, A)

^{is the}

type ^(I) (resp. (II))

^{in Theorem}

1.5,

then we

get

^the

type (2-2) (resp. ^(2-3))

in Theorem 1.6. This

completes

the

proof

of Theorem 1.6.

DEFINITION 1.8. Let X be an abelian surface and let

L, L1, ... , Ln

be

ample

line bundles ^on X.

(A)

^If

(X, L)

^{is the}

type ^(2-1) (resp. (2-2), (2-3))

ⁱⁿ Theorem 1.6 with

L 2 ; 2 a,

^{then we} say that

(X, L)

^{is the}

type (I; ^a) (resp. (II; a), (III; a)).

(B)

^{If X =}

J(B)

^and

Li

^{= B for a smooth curve B of} genus two and i

= 1, ... ,

n, then we call

(X, L1, ... , Ln )

^the

type (J),

where * denotes numerical

equivalence.

(0

^{If X =}

El

^X

E2

^{and L =}

pt(CD1) ® ~2 ( ~2 )

^with

(deg(JJ1, ( a , b)

^or

(deg 6D,, ( b , a),

^{then we call}

(X, L )

^the

type

(P; {a, 6}),

^where

^{El and E2}

^{are smooth}

^elliptic

^{curves, pi} is the i-th _pro-

jection,

^{and for i =}

1,

^2.

(D) If X = El

^X

E2

^and

Li

^{= with}

deg Di

⁼ ai , then we

call

(X, L)

^the

type (PS; t;

aI, ... ,

an ),

^where

El

^and

E2

^{are smooth}

ellip-

tic _{curves, pt} is the t-th

projection, Si

^{is a} section of _pt, and ~2 ^{E Pic}

(Et)

for

i = 1, ... , n.

2. The case in which

L1 + ... + Lt

^{is not}

ample

for t = k + 1 or 1~ . THEOREM 2.1. Let X be an abelian

surface,

^{and let}

L1

^and

L2 be ample

line bundles on X. We

put

^{L : =}

L1

⁺

L2 .

^{Then L is} not very

ample if

^and

only if (X, L1, L2 ) satisfies

^one

of

^the

foLLowing:

(1) (X,

^the

type (J),

(2) (X , L1, L2 )

^{is the}

type

^with a1 &#x3E;

0,

a2 &#x3E;

0,

^and

t = 1, 2.

PROOF. First we prove the «if»

part.

If

(X, L1, L2 )

^{is the}

type (1),

^then

^{L 2}

⁼

(L1

⁺

L2 )2

^{= 8 and}

h ° (L ) _

= L 2 /2

^{= 4. If L is very}

ample,

^{then X is a}

hypersurface

^of

degree

^{8 in}

^p3.

But this is

impossible

^{because X is an} abelian surface. Hence L is not very

ample.

If

(X, L1, L2 )

^{is the}

type (2),

^then

LF,

⁼

2,

^where

F1

^{is a fiber of} pi . If L is _very

ample,

^then

F1

⁼ But this is

impossible

^{because X is an}

abelian surface.

Next we prove the

«only

^if»

part.

^{Assume that L is} not very

ample.

Then

L 2

⁼

(L1

⁺

L2 )2 ~

^{8 because}

2, 2,

^{and 2.}

(A)

^{The case in which}

L 2

⁼ 8.

Then

L1 L2

^{= 2 and}

L2

^{= 2.}

By Hodge

^index

Theorem,

^{we obtain}

that

Li = L2.

^Since

g(L1 )

⁼

2,

^we

get

^that

(X, L1 )

^{is the}

type ^(I)

^or

^(II)

ⁱⁿ

Theorem 1.5.

If

(X, L1 )

^{is the}

type (I),

^then

L2

^{--- B.}

If

(X, L1 )

^{is the}

type (II),

^{then we can}

easily

prove

L2=pi*(Di0

®p2*D’2,

^where

6D! E Pic (Ei )

^with

deg D’1

⁼

deg 6D2’

^{= 1.}

(B)

^{The case in which}

L 2 ;

10.

By

^Reider’s

Theorem,

there exists an effective divisor D on X such that

(LD , D 2) = (2, 0) or (1, 0)

because the value of

D 2

is even. Since L ⁼

L1

⁺

L2

^and

Li

^is

ample

^{for i =}

1, 2,

^we

get (LD , D 2 )

⁼

( 2 , 0 ).

^{So we}

have

Li D

^{= 1} for each i.

By

^Remark

1.3.1,

there exists a fiber _space

f :

^X- C such that C is a smooth

elliptic ^{curve, X}

^{= C x}

F, f

is identified with the first

projection

^{via this}

isomorphism,

^{and D is a fiber}

of f,

^where

F is a fiber

of f.

^Since

Li F =

^{1 for each}

i,

^we obtain that

0 ^{for i =}

1, 2,

where

mj

^{E Pic}

(C) and Si

^{is a section}

of f.

^Since

Li

^is

ample,

^we

get Li Si

&#x3E; 0. Hence

deg 6Di

&#x3E; 0 because

Si2

^{= 0. This}

completes

the

proof

of Theorem 2.1.

COROLLARY 2.1.1. Let X be an abelian

surface

and let L be an am-

ple

line bundle on X. Then 2L is _{not very}

ample if

^and

only if (X, L) satisfies

^one

of

^the

following:

(1) (X, L) is

^the

type (J),

(2) (X, L) is

^the

type (P; ~ a , b 1)

^{with a} &#x3E; 0 and b = 1.

PROOF. We

put L1=

^{L and}

L2

^{= L. Then we}

get

the above result

by

Theorem 2.1. We remark that if then we use Lemma 1.3.

THEOREM 2.2. Let X be an abetian

surface

^{and let}

L1, ... , Lk

^{+ 1 be}

ample

line bundles on X. Assume that k ~ 2. We

put

^{L : =}

L1

^{+ ... +}

+

Lk

+ 1. Then L is not

k-very ample if and onLy if (X, L1,

... ,

Lk + 1 )

^{is the}

type (PS; t;

... ,

ak + 1 )

^{with ai} &#x3E; 0

for any i,

^{and t =}

1,

^2.

PROOF. First we prove the «if»

part.

^Let

Fl

^{be a fiber of Then}

+ 1. But

by

^Theorem

^1.4,

^{L is not}

k-very ample.

Next we prove the

«only

^if»

part.

^Then

Assume that L is not

k-very ample.

^Then

by

^Theorem

1.4,

there exists an

effective divisor D such that

Because LD ⁼

(L1

^{+ ...}

+ Lk + 1 ) D,

there exists an

ample

line bundle

Li

such that

Li D

^{= 1. We} may ^assume that i = 1.

By

^Remark

1.3.1,

^there exists a fiber

space f :

X - C such that C is a smooth

elliptic

curve, X =

= C ^x

F, f is

identified with the first

projection

^{via this}

isomorphism,

^{and D}

is a fiber

of f,

^{where F is a fiber}

of f.

On the other

hand, by

^Theorem

1.4,

we

get

^{LD ~ k + 1 since}

g(D)

^{= 1. Hence}

Li D

^{= 1 for} any i. Since D is a

fiber

of f and

&#x3E; 0 for i ⁼

1,

... , k +

1,

^we

get

^that

Li == f* CDi Q9

Q9 for i ⁼

1, ... , k

⁺

1,

where

6Di

^{E Pic}

( C), Si

^{is a section}

of f.

^Since

Li

is

ample,

^we

get Li Si

&#x3E; 0. Hence

deg 6Di

&#x3E; 0 because

S 2

^{= 0 for} any i. This

completes

^the

proof

of Theorem 2.2.

By

^Theorem

2.2,

^{we can} prove the

following Corollary.

COROLLARY 2.2.1. Let X be an abelian

surface

and let L be an am-

ple

line bundle on X Then

for

^{k ~}

^{2, (k}

⁺

^{1 ) L}

^{is not}

k-very ample if

^and

onLy if (X, L) is

^the

type (P; {o, b 1)

^{with a} &#x3E; 0 and b ⁼ 1.

THEOREM 2.3. Let X be an abelian

surface

^{and let}

L1,

^{... ,}

Lk be

ample

line bundles on X. Assume that k a 3. We

put L : = L1 + ... + Lk.

Then L is not

k-very ample if

^and

only if (X, L1, ... , Lk )

^{is one}

of

^the

following:

(1) k

^{= 3 and}

(X, L 1, L2 , L3 ) is

^the

type (J),

(2) (X, L1,

^{... ,}

Lk - 1 )

^{is the}

type (PS; ^t;

ai , ... , ak _

1 ),

^and

Lk

⁼

= put*

^{0 where}

6Dk

^{E Pic}

(Et )

^with

deg

6Dk

0,

pt is the t-th _pro-

jection,

^{T is a divisor on X with}

TFt

^{= 2}

for

^a

fiber Ft of

pt, ai is a

positi-

ve

integer for

ⁱ

= 1,

... , k -

1,

^{and t =}

1, 2,

(3)

^the

type (PS; t;

aI,

... , ak )

^with ai &#x3E; 0

for

...,

k,

^and

t=1, 2.

PROOF. First we prove the «if»

part.

If

(X, L1,

^{... ,}

Lk)

^{is the}

type ⁽¹⁾

in Theorem

2.3,

then k ⁼

3,

^{LB =}

- (L1

⁺

L2

⁺

L3) B

⁼

6, and g(B)

^{= 2. Then}

by

^Theorem

1.4, L

^is

not 3-very ample.

If

(X, L1,

^{... ,}

Lk )

^{is the}

type (2)

^or

(3)

in Theorem

2.3,

^then

~ k + 1 for a

fiber F1

^of pi. Then

by

^Theorem

1.4,

^{L is not}

k-very ample.

Next we prove the

«only

^if»

part.

We calculate

L 2 :

Since 1~ ~

3,

^we

get 2 k 2 ~ 41~ + 6.

^{So we have}

L2&#x3E;-

4 k + 6. Assume that L is not

k-very ample.

^Then

by

^Theorem

^1.4,

there exists an effective divi-

sor D on X such that

We _may assume that

Then we obtain that 2.

By Hodge

^index

^Theorem,

^we

get D 2 2.

(2.3.1 )

^{The case in which}

D 2 =

2.

Then

L1 D

^{= 2 and}

L1=

^D

by Hodge

index Theorem. Since

g(L1 )

⁼

^2,

we obtain that

(X, L1 )

^{is the}

type (I)

^or

(II)

in Theorem 1.5. On the other

hand, by g(D) = 2, L1 D = 2,

and Theorem

1.4,

^we

get

Hence k ⁼ 3 and

L1 D

⁼

L2 D

⁼

L3 D

^{= 2. So}

by Hodge

^index

Theorem,

we

get L1 = L2 = Z.3

= D. Therefore we

get

^the

type ⁽¹⁾

^or

⁽³⁾

^{in Theorem}

2.3.

(2.3.2)

^{The case in which}

D 2

⁼ 0.

(2.3.2.1)

^{The case in which}

L1 D

^{= 2.}

Then

by assumption,

^we have LD ~ 2 k. Hence

by

^Theorem

^1.4,

^we

get

2 k ~ LD ~

2 g(D )

+ 1~ - 1 = l~ + 1. Therefore k 5 1 and this is a contra- diction.

(2.3.2.2)

^{The case} in which

L1 D

^{= 1.}

By

^Remark

1.3.1,

there exists a fiber

space f :

X- C such that C is a

smooth

elliptic curve, X =

^{C x}

F, f

^is identified with the first

projection

via this

isomorphism,

^{and D is a fiber}

of f,

where F is a fiber

of f.

^Since

k~(L1+...+Lk)D~2g(D)+k-1 =k+1,

^we

get

type (2)

in Theorem

2.3,

and if

(L1 D , ... , Lk _ 1 D , Lk D ) _ ( 1,

^{... ,}

1, 1),

then

(X, L1,

... ,

Lk)

^{is the}

type (3)

in Theorem 2.3

by using

^{the same ar-}

gument

^{as in the}

proof

of the above Theorems. This

completes

^the

proof

of Theorem 2.3.

THEOREM 2.4. Let X be an abelian

surface

and let L be an

ample

Line bundles on X. Then

for k ~ ^2,

^{kL is not}

k-very ample if

^and

only if (X, L)

^{is one}

of

^the

following:

(1) k

^{= 2 or}

3,

^and

(X, L) is

^the

type (J),

(2) (X, L)

^is the

type (P; {a, b 1)

with a &#x3E; 0 and b = 1.

PROOF. For k a

3,

^{this is a}

corollary

of Theorem 2.3. Assume that k ⁼ 2.

By

^{the same}

argument

^{as in the}

proof

of Theorem

2.3,

^{we can}

prove the «if»

part.

^{So we} prove the

«only

^if»

part.

Assume that L is not

2-very ample.

(2.4.1)

^{The case in which}

L 2 ~ 4.

Then

( 2 L )2 ~ 16,

^and

by

^Theorem

^1.4,

there exists an effective divisor D on X such that

(2L)

^{D ~}

2g(D)

^{+ 1 ~ 5. If}

^{D 2}

&#x3E;

0,

then

by Hodge

^in-

dex

Theorem,

^we

get

^{LD ~ 3} and this is

impossible.

^Hence

^{D 2 =}

^{0 and}

g(D)

^{= 1.} Therefore 2 LD ~

3,

^that

is,

^{we obtain LD = 1.} So D is a smooth

elliptic

^curve.

By

Lemma 1.2 and Lemma

1.3,

^{we can} prove that

(X, L )

^is the

type (P; ~ a , b 1)

^with a &#x3E; 0 and b = 1.

(2.4.2)

^{The case in which}

L 2 =

2.

Then

(X, L)

^{is one} of the

type

in Theorem 1.5.

This

completes

^the

proof

of Theorem 2.4.

3. The case in which

(n - t) L

^{is not}

k-very ample

^{for t =1 and 2.}

THEOREM 3.1. Let

(X, L) be

^a

polarized

^abelian

surface.

^Let

(k, t)

be a

pair of integer

^which

satisfies

^the

following inequalities;

(A) k ; 3t + 1, (B) t ~

^1.

Then

( k - t)L

^{is not}

k-very ample if

^and

only if

^one

of

^the

following types holds;

(1) (k, t)

⁼

(4, 1), (5, 1 ),

^or

(7,2),

^and

(X, L) is

^{the the}

type (J), (2) (X, L)

^{is the}

type (P; {a, b 1)

^{with a} &#x3E; 0 and b = 1.

PROOF. First we remark that

( k - t )2 L 2 ;

^{4 k + 6 unless}

(L 2 , k , t )

⁼

- (2, 4, 1 ).

We _prove the «if»

part

of Theorem 3.1. If

(X, L)

^{is the}

type (1)

ⁱⁿ Theorem

3.1,

^then

( k - t)

^{LB ~}

2g(B)

^{+ k - 1 ~ 2 k + 1. Hence}

by

^Theo-

rem

1.4,

^{L is not}

k-very ample.

^If

(X, L)

^{is the}

type (2)

in Theorem

3.1,

then

(k - 2g(Fl )

⁺ k - 1 ~ 2k + 1 for _any fiber

F1

^of Pl. Hence L is not

k-very ample by

Theorem 1.4.

Next we prove the

«only

^if»

part.

(3.1.1 )

The case in which

L 2 ~

4.

Since

(k - t)2 L 2 ~ 4 k + 6, by

^Theorem

^1.4,

there exists an effective divisor D on X such that

( k - t)

^{LD ~}

2 g(D )

^{+ k - 1 ~ 2 k + 1.}

Assume that

D 2

&#x3E; 0.

Then, by Hodge

^index

Theorem,

^we

get

^{LD ~ 3.}

Hence

3(k-t)~(k-t)LD~2k+1,

^{and we} obtain k~3t+1.

By hy- pothesis

^{we have k = 3 t + 1 and LD = 3. So we}

get ^{D 2}

^{= 2 and}

g(D)

^{= 2.}

By

^Theorem

1.4,

^{we have}

3( 2 t

⁺

1)

⁼

(k - t)

^{LD ~ k + 3 =}

= 3 t + 4,

^{and so we} have t 1 /3. But this is a contradiction.

Hence

D 2

⁼ 0 and

g(D)

^{= 1.}

By

^Theorem

1.4,

^{we obtain}

( k - t ) LD ~

~k+ 1.

Assume that LD ~ 2. Then we

get

k - 1 ~ 2 t. Since

k ~ 3 t + 1,

we

get

3 t ~ 2 t. This is

impossible

^{because t ~ 1. Hence LD = 1.}

Therefore D is a smooth

elliptic

^{curve and}

by

^{the same method as in}

the

proof

of the above

theorems,

^we

get

^that

(X, L)

^{is the}

type

(P; {a,

^with a &#x3E; 0 and b ⁼ 1.

(3.1.2)

The case in which

L 2 = 2.

Then

by

^Theorem

1.5,

^we

get

^that

(X, L )

^{is one} of the

types

^{in Theo-}

rem 1.5.

In order _{to prove} Theorem

3.1,

^{it is} sufficient to

study

^{the case in}

which X ⁼

J(B)

^{and L =}

B,

where B is a smooth

projective

^{curve of} genus 2 and

J(B)

^is

its jacobian variety.

^If

( k , t)

⁼

(4, 1),

^then

( k - t )2 L 2

4 k + 6 and

( k - t ) L

^is not

k-very ample by

^{Theorem 1.4. So we assume}

that

( k , t ) ~ ( 4 , 1 ).

^Then

( k - t )2 L 2 ~ 4 k + 6.

^Then

_by

^Theorem

1.4,

there exists an effective divisor D on X such that

( k - t)

^{LD ~}

2g(D)

⁺

+ k - 1 ~ 2 k + 1. Here ^we remark that we can prove LD ~ 2

by

^Remark

1.3.1 since L * B.

(3.1.2.1)

^{The case in} which LD ~ 3.

Then 3k-3t;

(k-t)LD~2k+1.

^{So we} have k~3t+1.

By hy- pothesis,

^we

get

^{k = 3 t + 1 and LD = 3. Since}

^{L 2}

⁼

2,

^{we obtain}

D 2 ;

4.

Hence2k-5 ~3t. and 4. Be-

cause k ⁼ 3 t

+ 1,

^we obtain

(k, t)

⁼

(4, 1 )

and this is a contradiction.

(3.1.2.2)

The case in which LD ⁼ 2.

If

D~

⁼

0,

^then

g(D)

^{= 1. So}

by

^Theorem

1.4,

^we

get 2(k - ^{t) _}

= (k - t)

^{LD ~ k + 1. Hence k - 1 ~ 2 t.} Since k ~ 3 t +

1,

^{this is}

impossi-

ble because t ~ 1.

If

D 2

&#x3E;

0,

^{then L =} D since LD ⁼ 2 and

L 2

= 2.

By

^Theorem

1.4,

^we

get 2(k-t) _ (k-t)LD~k+3.

Hence k-3 ~2t. Since

k~3t+1,

^we

get

^{3 t - 2 ~ 2 t.} Therefore t S 2.

If

t = 1,

^then

2(k -1 ) _ (k - t) LD ~ k + 3.

Hence k ~ 5. Because

4 = 3 t + 1; k,

^we

get

^{k = 5}

by

^the

assumption

^that

(k, t) # (4, 1 ).

If

t = 2,

^then

2(k - 2 ) _ (k - t) LD ~ k + 3.

Hence k ~ 7. Because

7=3t+1 ;k,

^we

get

^k=7.

This

completes

^the

proof

of Theorem 3.1.

THEOREM 3.2. Let

(X, L)

^{be a}

polarized

^abetian

surface.

^Let

( k , t)

be a

pair of integers

with 3 t ; k ~ 2 t +

1,

^{t ~}

1,

^and

( k , t ) ~ ( 3 , 1 ).

^Then

(k - t)

^{L is not}

k-very ample if

^and

only if

^one

of

^the

following

holds:

and

tg ~ 3,

^and

(X, L) is

^{the the}

type (J),

(2) (X, L) is

^the

type (P; {a, b ~ )

^{with a ~ 1 and}

^{b = 1,}

(3) k = 2 t + 1

^and

(X , L )

^{is the}

type (P; ~ a , b 1)

with a ~ 1 and

b = 2,

(4) k

^{= 2 t + 1 and}

(X, ^L)

^{is one of the}

type (I; 2), (II; 2),

^or

(III; 2).

PROOF. First we prove the «if»

part.

^If

(X, L)

is the

type ⁽¹⁾

^{in Theo-}

rem

3.2,

^then

( k - t )

^{LB ~}

2 g(B )

^{+ k - 1 ~} 2 k. Hence L is not

k-very

^am-

ple by

^{Theorem 1.4. If}

^{(X, L)}

^{is the}

type ⁽²⁾

^or

⁽³⁾

in Theorem

3.2,

^then

(k - t) 2g(Fl)

^{+ k - 1 ~ 2k for a fiber}

F1

^of pl. Hence L is not k- very

ample by

^{Theorem 1.4. If}

(X, L)

^{is the}

type (4)

^{in Theorem}

3.2,

^then

(k - t )

^{LF ~}

2 g(F)

+ k - 1 ; 2 k for a fiber F

of f.

Hence L is not

k-very ample by

Theorem 1.4.

Next we prove the

«only

^if»

part.

(A)

^{The case in which}

L 2 ~

4.

First we remark that

(k - t)2 L 2 ;

^{4 k + 6}

by assumption.

^Hence

by

Theorem 1.4 there exists an effective divisor D on X such that

(k - t) LD 2g(D) + k -1 2k + 1.

If LD ~

4,

^then

4(k - t) ~

^{2 k +}

1,

^that

is,

^{2 k ~ 4 t + 1.} Since k ~ 2 t +

+ 1,

^we

get

that 4 t + 1 ; 2 k ~ 4 t + 2. But this is

impossible.

^Hence

LD~3.

If

LD = 3,

^then

D 2 ~ 2

since

L 2 ~ 4.

Therefore

3(k - t) =

= (k - t) LD £ k + 3 because g(D ) ~ 2.

^{So we have} 2 k ~ 3 t + 3.

Since k ; 2 t + 1,

we obtain that 3 t + 3 ~ 4 t + 2. Since t ~

1,

^{we have t = 1 and k = 3. But}

by assumption

^{this is a} contradiction. Hence LD ~ 2 and

D 2

⁼ 0.

If LD ⁼

1,

^{or LD = 2} such that D is not an irreducible reduced _curve, then there exists ^an irreducible reduced curve B such that LB = 1. Then

by

^{Lemma 1.2 and Lemma}

1.3,

^we

get

^the

type ⁽²⁾

in Theorem 3.2.

If LD ⁼ 2 such that D is an irreducible curve, then D is a smooth el-

liptic

^{curve. Then}

^{2(k - t )}

⁼

^{( k - t )}

^{LD ~}

2 g(D )

+ k - 1 = k + 1. Hence k ~ 2 t + 1.

By assumption

^{k = 2 t + 1 in this case. Then}

by

^Theorem

^1.6,

we

get

^the

type (3)

^or

(4)

^{in Theorem 3.2.}

(B)

^{The case} in which

L 2 =

2.

Then

(X, L)

^{is the}

type (I)

^or

(II)

ⁱⁿ Theorem 1.5. If

(X, L)

^{is the}

type (II)

^{in Theorem}

1.5,

^then

(X, L)

^{is the}

type (2)

in Theorem 3.2. So it is sufficient to

study

^{the case in which}

(X, L )

is the

type ^(I)

^{in Theorem}

1.5.

Assume that X =

J(B)

^{and L} is the class of a translation of the theta

divisor,

where B is a smooth

projective

^{curve of genus 2 and}

^J(B)

^{is its}

jacobian variety.

^{Then we} remark that

( k - t )2 L 2 ~ 4 k + 6 _(resp.

( k - t )2 L 2 4 k + 6)

^{if t ~ 4}

(resp. t ~ 3).

^If

t ~ 3,

^{then we}

get

^that

( k - t ) L

is not

k-very ample by

Theorem 1.4. Since 3 t ~ k ; 2 t + 1 and

(k, t ) ~

~

( 3 , 1 ),

^we

get

^that

( k , t)

⁼

(9, 3),(8, 3),(7, 3),(6, 2)

^or

( 5 , 2 ).

Assume that t ~ 4. Then

by

^Theorem

^1.4,

there exists an effective di- visor D on X such that

( k - t)

^{LD ~}

2 g(D )

⁺ k -1 ~ 2 k + 1.

IfLD ~

4,

^{then this is}

impossible by

^{the same}

argument

^{as in the case}

(A).

Hence LD ~ 3. We remark that LD ~ 2 since L is the class of a

translation of the theta divisor. Since

L 2 = 2,

^we

get

^that

^{D 2 ~ 4}

^and

g(D) ~

^3.

If LD ⁼

3,

^then

3(k - t ) ~ k

⁺ 5. Hence 2 k ~ 3 t + 5. Since

k ; 2 t + 1,

we

get 4 t + 2 ~ 2 k ~ 3 t + 5,

^that

is,

t ~ 3. But this is a contradiction.

If LD ⁼

2,

^then

D 2-

2.

If D 2= 2,

^then

g(D)

^{= 2 and}

2(k - ~) ~

^{+ 3.}

Thus we have k ~ 2 t ⁺ 3. Hence k ⁼ 2 t +

1,

^{2 t +}

2,

^{and 2 t + 3.}

If

D 2

⁼

0,

^then

g(D)

^{= 1 and}

2(k - t) ~

^{k + 1. Hence k ~ 2 t + 1. Since}

k;2t+1,

^we

get k=2t+1.

This

completes

^the

proof

of Theorem 3.2.

COROLLARY 3.3. Let

(X, L)

^{be a}

polarized

^abelian

surface.

^Assume

that k a 2 t +

4 for ( k , t)

^E

~T®2.

Then

( k - t)

^{L is not}

k-very if

^and

only if (X, L)

^{is the}

type (P; ~ a , b 1)

^{with a} &#x3E; 0 and b =1.

PROOF. This is obtained

by

Theorem 3.1 and Theorem 3.2.

THEOREM 3.4. Let

(X, L) be

^a

polarized

^abelian

surface.

^Assume

with k ~ 3. Then

(k - 1 )L is

^not

k-very ample if

^and

only if

one

of

^the

following

^holds:

( 1 )

^{k =}

3, 4,

^or

5,

^and

(X, L)

^{is the}

type (J),

(2) (X, L)

^{is the}

type (P; I a,, a2 1)

^{with a1} &#x3E; 0

and a2

⁼

1,

(3) k = 3

^and

(X, L) is

^the

type b2 ~ )

^{with and}

b2=2,

(4)

^{k = 3 and}

(X, L) is

^one

of

^the

type (1; 2), (II; 2),

^or

(III; 2),

(5)

^{k=3 and}

L2=4.

PROOF. We ^can

easily

prove the «if»

part by

^{Theorem 1.4. So we}

prove the

«only

^if»

part.

^{If k ~}

4,

then this is a

corollary

of Theorem

3.1,

and

(X, L)

^{is one of the}

type ⁽¹⁾

^or

(2)

^{in Theorem 3.4. So we} may ^assume that k ⁼ 3.

(A)

^{The case in which}

L 2 ;

6.

Then

4 L 2

⁼

( k -1 )2 L 2 ;18

^{= 4 k + 6. Hence}

by

^Theorem

1.4,

^there exists an effective divisor D on X such that 2 LD ~

2g(D)

+ 2 ~ 7. Hence LD 3. Since

L 2 ~

6 we

get

^that

^{D 2}

^{= 0 and}

g(D )

⁼ 1. Therefore 2 LD ~

~ 4,

^that

is,

^{LD ~ 2.}

If LD ⁼

1,

^{or LD = 2 and D is not} irreducible and

reduced,

^{then there}

exists an irreducible and reduced curve B on X such that LB ⁼ 1. Hence

by

Lemma 1.2 and Lemma

1.3,

^we

get

^that

(X, L)

is the

type ⁽²⁾

^{in Theo-}

rem 3.4.

If LD ⁼ 2 and D is irreducible and

reduced,

^then

by

Theorem 1.6 we

get

^that

(X, L)

^{is one of the}

type (3)

^or

(4)

in Theorem 3.4.

(B)

^{The case in which}

L 2

⁼ 2.

Then

by

^Theorem

^1.5,

^we

^get

^that

(X, L )

^{is one of the}

type (1)

^or

(2)

ⁱⁿ Theorem 3.4.

(0

^{The case in which}

L 2

⁼ 4.

Then

(X, L)

^{is the}

type (5)

in Theorem 3.4.

This

completes

^the

proof

of Theorem 3.4.

THEOREM 3.5. Let

(X, L)

^{be a}

polarized

^abelian

surface.

^Assume

with k ~ 4. Then

(k - 2 ) L

^{is not}

k-very ample if

^and

only if

one

of

^the

following

^holds:

( 1 )

^{k =}

4 , 5 , 6,

^or

7,

^and

(X, L)

^{is the}

type (J),

(2) (X, L)

^{is the}

type (P; I a,, a2 1)

^{with a1} &#x3E; 0

and ct2

⁼

1,

(3) k

^{= 4 or}

5,

^and

(X, ^L)

^{is the}

type b2 ~ )

^{with and}

b2

⁼

2,

(4) k

^{= 4 or}

5,

^and

(X, L)

^{is one}

of

^the

type (I; 2), (II; 2),

^or

(III; 2),

(5)

^{k=4 and}

L2=4.

PROOF. We can

easily

prove the «if»

part by

Theorem 1.4. So we

prove the

«only

^if»

part.

^{If k ;}

7,

then this is a

corollary

of Theorem

3.1,

and

(X, L)

^{is one of the}

type ⁽¹⁾

^or

⁽²⁾

in Theorem 3.5.